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We study locally compact groups for which the Fourier algebra coincides withthe Rajchman algebra. In particular, we show that there exist uncountably manynon-compact groups with this property. Generalizing a result of Hewitt andZuckerman, we show that no non-compact nilpotent group has this property,whereas non-compact solvable groups with this property are known to exist.  We provide several structural results on groups whose Fourier and Rajchmanalgebras coincide as well as new criteria for establishing this property.  Finally, we study the relation between groups with completely reducibleregular representation and groups whose Fourier and Rajchman algebras coincide.For unimodular groups with completely reducible regular representation, we showthat the Fourier algebra may in general be strictly smaller than the Rajchmanalgebra.

Groups whose Fourier algebra and Rajchman algebra coincide